Investigation of the behavior of vertical fracture penetration/deflection at the interface between coal seam and mudstone layers

Abstract

Based on the energy release rate criterion, the behavior of vertical fracture penetration/deflection at interface between coal seam and mudstone layers are investigated by finite element method (FEM) in this study. Considering its interfacial effect that is simulated by linear spring model (LSM), the energy release rates (ERRs) of vertical fracture penetration and deflection at the interface are calculated using virtual crack close technique (VCCT). Combined with the critical energy release rates (CERRs) of coal seam layer, mudstone layer, and sandstone layer as well as the interface between them were tested in a laboratory. The layers were gathered from the southern Qinshui Basin. The judging methods of vertical fracture penetration/deflection at interface are presented and the maximum penetration length which extends into the mudstone layer is calculated. The result is consistent with the analyses from the field monitoring results. Furthermore, the effects of interfacial stiffness, coal seam thickness, Young’s modulus of sandstone cap and in-situ differential stress on the propagation behavior are also evaluated. The results show that the interfacial stiffness of interface between coal seam and mudstone layers vary between 10⁹ Pa/m and 10¹² Pa/m, which is about 10¹⁰ Pa/m. The state of interface is separation or perfect when the interfacial stiffness is smaller than 10⁹ Pa/m or larger than 10¹² Pa/m. With the increase of coal seam thickness, the maximum penetration length gradually and faintly decreases. In contrast, the maximum penetration length obviously decreases when Young’s modulus of sandstone cap raises or the in-situ differential stress Δ𝜎 declines. In other words, Young’s modulus of sandstone cap and the in-situ differential stress Δ𝜎 are the main factors which influence the behavior of vertical fracture penetration/deflection at the interface between coal seam and mudstone layers.

Keywords

Interfacial effect energy release rate interfacial stiffness linear spring model virtual crack closure technique

1. Introduction

Coalbed methane (CBM), which is unconventional natural gas resources adsorbed to the coal reservoir, is a type of clean fuel in recent years. China has abundant CBM resources, especially in the southern Qinshui Basin (the proved reserves is 3.95 × 10¹² m³), which is an important CBM gathering area of China [22]. As the low permeability of coal reservoir, most CBM wells must be stimulated by hydraulic fracturing that will generate series of fractures especially the type of vertical fracture [10,15]. When the vertical fracture propagates to the interface between coal seam and mudstone layers, there are three pathways to choice from, which are stopping extension, extending along formation interface and directly penetrating into bounding layer [19]. The direction of the vertical fracture and how long the vertical fracture penetrates into the mudstone are very important to measure the CBM production [1].

In view of the propagation behavior of vertical fracture at material interface, Biot et al. [3] assumed the perfect interface between formations and neglected the leak- off of fracturing fluid. Then a simple criterion is introduced without considering the in-situ stress difference $\begin{eqnarray}\displaystyle \displaystyle \frac{p_{1}}{p_{2}}=\displaystyle \sqrt{\frac{S_{2}G_{2}}{S_{1}G_{1}}} & & \displaystyle\end{eqnarray}$ (1) where p ₁ and p ₂ are extending pressure in productive layer and bounding layer respectively, G ₁ and G ₂ are shear modulus of productive layer and bounding layer respectively, S ₁ and S ₂ are interface energy density of productive layer and bounding layer respectively. If S ₁ G ₁ > S ₂ G ₂, the fracture will propagate into bounding layer. Otherwise, the fracture will stop extending.

Considering in-situ stress 𝜎_xx and 𝜎_yy, rock tensile strength T ₀ and interfacial friction coefficient 𝜇, Renshaw and Pollard [14] put forward a criterion which can be expressed as $\begin{eqnarray}\displaystyle \displaystyle \frac{-{\sigma}_{yy}}{T_{0}-{\sigma}_{xx}}>\frac{0.35+0.35/{\mu}}{1.06} & & \displaystyle\end{eqnarray}$ (2) If the condition of Eq. (2) is satisfactory, the fracture will propagate into bounding layer, otherwise it will deflect into the interface. Zhao and Chen [19] as well as Liu and Wang [11] analysed the fracture propagation at interface in view of stress-strain theory. However, due to the singularity of stress field at the tip of fracture, it has many limits and shortages to use stress criterion to judge the crack propagation paths. An increasing number of scholars [2,4,8,16] utilised the VCCT framework, which can avoid the stress singularity without using stress field around a crack tip, to calculate stress intensity factor or energy release rate to crack propagation judging. Unfortunately, these studies assume the interface between two materials with a perfect state which does not match the actual situation. In order to describe the imperfection of interface between materials, Zhong et al. [21], Vladislav et al. [17], and Gao et al. [5] have used linear spring model to simulate the material interface. In the field of petroleum engineering, Yang et al. [18] has used linear spring model to simulate the first interface between casing and cementing and second interface between cementing and formation. Peirce et al. [13] and Gu et al. [6] have adopted linear spring model to simulate the interface between the layered formation.

The present work provides a VCCT framework that allows for a simulation of interfaces with LSM between coal seam and mudstone layers and mudstone and sandstorm layers. Using the User Subroutine UVERRS, a calculation method of ERRs is embedded in the Abaqus VCCT framework. Furthermore, using the three point bending test, the CERRs of coal seam, mudstone, and sandstone layers as well as the interface between them are tested and predicted. The energy release rate criterion is then used to investigate the behavior of vertical fracture penetration/deflection at the interface between coal seam layer and mudstone layer.

2. Model description

The No. 3 coal seam of Shanxi Formation of Permian in the southern Qinshui Basin is one of the primary mineable coal beds. According to Kavanagh and Pavie [12], the thickness of No. 3 coal seam which is buried underground of 500--600 m is usually between 5--7 m with an average of 6 m. The upper and lower interlayer is the mudstone of 10 m thick with lower mechanical strength. No. 3 coal seam layer and mudstone layer are the fracturing layers which are constrained by tight sandstone cap. The average pore pressure, maximum horizontal stress and minimum horizontal stress of No. 3 coal reservoir are 4.8 MPa, 9.9 MPa and 7.5 MPa, respectively. The porosity and the permeability is 5% and 2 md, respectively.

In order to simulate the actual situation, the cores of coal seam layer, mudstone layer and sandstone layer are gathered from the southern Qinshui Basin to test the mechanical parameters in a laboratory (Table 1). The mechanical parameters such as Young’s modulus E, Poisson’s ratio 𝜈 and compressive strength 𝜎_b are carried out through a series of unsaturated triaxial compression tests. The fracture toughness K _C is tested by the three point bending technique and then the CERR is calculated. According to Kavanagh and Pavier [9], the fracture toughness of interface between two materials is between the two. Hence, the CERRs of interfaces between coal seam and mudstone layers and mudstone and sandstone layers this study utilized are approximately set as 350 N/m and 420 N/m, respectively.

Table 1
The mechanical parameters of coal seam layer, mudstone layer and sandstone layer

Property Coal seam layer Mudstone layer Sandstone layer

Young’s modulus E (GPa) 4.73 6.56 34.5

Poisson’s ratio 𝜈 0.20 0.27 0.30

Compressive strength 𝜎_b (MPa) 16 27 39

Fracture toughness K _C (MPa/m^1∕2) 1.20 1.58 4.05

Critical energy release rate G _C (N/m) 304.44 380.55 475.43

Property	Coal seam layer	Mudstone layer	Sandstone layer
Young’s modulus E (GPa)	4.73	6.56	34.5
Poisson’s ratio 𝜈	0.20	0.27	0.30
Compressive strength 𝜎_b (MPa)	16	27	39
Fracture toughness K _C (MPa/m^1∕2)	1.20	1.58	4.05
Critical energy release rate G _C (N/m)	304.44	380.55	475.43

Note: the critical energy release rate G _C is calculated with the formula $G_{C}=K_{C}^{2}/E$ .

Considering a vertical fracture within the coal seam layer propagate at the interface between coal seam and mudstone layer, the geometry model shown in Fig. 1 is presented. The dip angles of formation composed of coal seam, mudstone and sandstone layer are assumed to be 0° in the model. In-situ stresses (composed of 𝜎_x in X-direction and 𝜎_z in Z-direction) are applied to the model as the boundary conditions. The vertical fracture in the model is always symmetrically generated and expanded along the direction of the maximum principal stress. The linear spring model is used to simulate the interfaces between coal seam and mudstone layers and mudstone and sandstone layers.

Fig. 1.

The geometry model.

3. Calculation of ERR by VCCT

According to the integral formula of crack closure suggested by Irwin [7], when the crack extends $\Delta$ a, the ERR is equal to the requiring energy the crack closed from a + $\Delta$ a to a. Thus, the formula for calculating the ERR of vertical fracture penetration/deflection at the interface can be expressed as $\begin{eqnarray}\displaystyle G=G_{I}+G_{II}, & & \displaystyle\end{eqnarray}$ (3) where G _I and G _II are the energy release rates corresponding to the tractions 𝜎 and 𝜏.

3.1. The ERR of vertical fracture penetration through the interface

When the vertival fracture penetrates through the interface, the ERR of vertical fracture along the Z-direction can be calculated as $\begin{eqnarray}\displaystyle G_{I}=\lim _{{\Delta}a\rightarrow 0}\displaystyle \frac{1}{2{\Delta}a}\int _{0}^{{\Delta}a}{\sigma}_{xx}{\Delta}vdz, & & \displaystyle\end{eqnarray}$ (4) and $\begin{eqnarray}\displaystyle G_{II}=\lim _{{\Delta}a\rightarrow 0}\displaystyle \frac{1}{2{\Delta}a}\int _{0}^{{\Delta}a}{\tau}_{zx}{\Delta}udz. & & \displaystyle\end{eqnarray}$ (5)

According to the definition of the fracture unit of four-node, an exemplary mesh with a fracture opening modeled by phantom node is depicted in Fig. 2. With the FEM of VCCT, the calculation process of ERR can be given as $\begin{eqnarray}\displaystyle G_{I}\cong \displaystyle \frac{F_{x1}{\Delta}u}{2{\Delta}a} & & \displaystyle\end{eqnarray}$ (6) and $\begin{eqnarray}\displaystyle G_{II}\cong \displaystyle \frac{F_{z1}{\Delta}w}{2{\Delta}a}, & & \displaystyle\end{eqnarray}$ (7) where Δu and Δw are the relative displacements of solid node 3 and solid node 4 in the local coordinate system respectively, and F _x1 and F _z1 are the nodal forces at the fracture tip as shown in Fig. 2.

Fig. 2.

The fracture unit of four-node. Solid and hollow circles illustrate the original and phantom nodes, respectively.

3.2. The ERR of vertical fracture deflection along the interface

If the vertical fracture deflects along the interface, the ERR of vertical fracture along the X-direction can be calculated as $\begin{eqnarray}\displaystyle G_{I}^{\prime }=\lim _{{\Delta}a\rightarrow 0}\displaystyle \frac{1}{2{\Delta}a}\int _{0}^{{\Delta}a}{\sigma}_{zz}{\Delta}vdx & & \displaystyle\end{eqnarray}$ (8) and $\begin{eqnarray}\displaystyle G_{II}^{\prime }=\lim _{{\Delta}a\rightarrow 0}\frac{1}{2{\Delta}a}\int _{0}^{{\Delta}a}{\tau}_{xz}{\Delta}udx. & & \displaystyle\end{eqnarray}$ (9)

According to the definition of the fracture unit of four-node, an exemplary mesh with a fracture opening along the interface modeled by phantom node is depicted in Fig. 3. With the FEM of VCCT, the calculation process of ERR of vertical fracture deflection along the interface can be given as $\begin{eqnarray}\displaystyle G_{I}^{\prime }\cong \displaystyle \frac{F_{z1}^{\prime }{\Delta}w^{\prime }}{2{\Delta}a}, & & \displaystyle\end{eqnarray}$ (10) and $\begin{eqnarray}\displaystyle G_{II}^{\prime }\cong \displaystyle \frac{F_{x1}^{\prime }{\Delta}u^{\prime }}{2{\Delta}a}, & & \displaystyle\end{eqnarray}$ (11) where Δu′ and Δw′ are the relative displacements of node 3 and node 4 in the local coordinate system respectively, and F′ _x1 and F′ _z1 are the nodal forces at the crack tip.

Fig. 3.

The fracture unit along the interface. Solid and hollow circles illustrate the original and phantom nodes, respectively. The linear spring creates a bond in both sides of the interface.

Considering the weak interface between coal seam and mudstone layers, and mudstone and sandstone layers, the linear spring model is used to simulate the interfacial effect between them. It is different from some investigations in recent years which used the linear spring model to simulate the interfacial shear slip [6,13]. The linear spring model is used to stimulate the interfacial shear slip and normal tensile of elastic layers in this paper as depicted in Fig. 4.

Fig. 4.

The linear spring model for stimulating interfacial shear slip (a) and normal tensile (b).

The elastic layers are connected at their interfaces by linear springs that satisfy the following conditions: $\begin{eqnarray}\displaystyle {\sigma}_{ij}={\beta}_{ij}{\Delta}u_{ij}={\beta}_{ij}\left(u_{ij}^{I}-u_{ij}^{II}\right), & & \displaystyle\end{eqnarray}$ (12) where 𝜎_ij and Δu _ij represent the jumps in the stress and displacement components across the interface respectively, 𝛽_ijis the stiffness constant associated with the interface, the subscript i and j represent X and Z directions respectively, the superscript I and II represent the two materials.

With the above formulas, the F _x1 and F _z1 in (8) and (9) can be respectively expressed as: $\begin{eqnarray}\displaystyle F_{x1}^{\prime }={\beta}_{xz}{\Delta}u_{xz}\quad \text{and}\quad F_{z1}^{\prime }={\beta}_{zz}{\Delta}u_{zz}. & & \displaystyle\end{eqnarray}$ (13)

4. The criteria of vertical fracture penetration/deflection at the interface between coal seam layer and mudstone layer

With the fracture criterion based on ERR, the necessary condition that a fracture penetrates across the interface into the next material can be expressed as: $\begin{eqnarray}\displaystyle \displaystyle \frac{{\psi}_{d}}{{\psi}_{p}}>\frac{G_{d}}{G_{p}}, & & \displaystyle\end{eqnarray}$ (14)where 𝛹_d and 𝛹_p respectively denote the critical energy release rate for fracture deflection along and penetration through the interface; G _d and G _p respectively denote the ERRs for crack deflection along and penetration through the interface, which will be calculated with the VCCT method in the paper.

Under the condition of Eq. (14), the fracture will penetrate through the interface, otherwise it will deflect along the interface with the reverse condition. Therefore, for the three-phase composed of coal seam, mudstone and sandstone layers, the condition of fracture penetration across interface can be expressed as: $\begin{eqnarray}\displaystyle \displaystyle \frac{{\psi}_{dc}}{{\psi}_{pm}}>\displaystyle \frac{G_{dc}}{G_{pm}}, & & \displaystyle\end{eqnarray}$ (15) when the fracture propagates at the interface between coal seam and mudstone layers, and $\begin{eqnarray}\displaystyle \displaystyle \frac{{\psi}_{dm}}{{\psi}_{ps}}>\displaystyle \frac{G_{dm}}{G_{ps}}, & & \displaystyle\end{eqnarray}$ (16) when the fracture propagates at the interface between mudstone and sandstone layers.

Where 𝛹_dc and 𝛹_dm are the critical energy release rate of the interface between coal seam and mudstone layers and mudstone and sandstone layers, respectively. 𝛹_pm and 𝛹_ps are the critical energy release rate of mudstone layer and sandstone layer, respectively. G _dc and G _dm denote the ERRs for crack deflection along the interface between coal seam and mudstone layers and mudstone and sandstone layers. G _pm and G _ps represent the ERRs of mudstone layer and sandstone layer.

5. The numerical simulation

5.1. Performance and verification of the numerical analysis

Based on the model description of the southern Qinshui Basin described in Section 2, a FEM model using the commercial finite element software Abaqus is performed, as is shown in Fig. 5. In the FEM model, the linear spring element is used to depict the interface behaviour. Two-dimensional models of the single lap joints using 4-node plane strain solids elements with the freedom of pore pressure (CPE4P from Abaqus library) for the computational meshes are implemented, which has 36000 elements. By calculating, it has less influence on results if the number of elements is bigger than 24000.

With the above FEM model the nodal forces (see Fig. 6) and nodal displacements (see Fig. 7) of the fracture tip can be calculated. Then the ERRs of vertical fracture penetration and deflection (G _pm and G _dc) at the interface can be carried out by the theory of VCCT with the data of nodal force and nodal displacement, which is developed to a user subroutine embedding into the ABAQUS. And the ERRs of G _pm and G _dc are 388.3 N/m and 25.2 N/m respectively. Using the datas from Table 1 and combined with the criterion formula of Eq. (14), the fracture will penetrate the interface and get into the mudstone layer with a certain length a _p. Furthermore, the ERRs of G _pm and G _dc are calculated with the increase of a _p (Fig. 8).

Fig. 5.

The FEM model by ABAQUS software.

Fig. 6.

Nodal forces of 𝜎_zz and 𝜏_xz distribution of the model.

Fig. 7.

Nodal displacements of u_x and u_z distribution of the model.

Fig. 8.

The ERRs with the increase of a _p.

It is noted that G _pm∕G _dc will decrease with the increase of a _p (Fig. 8). Meanwhile, G _pm∕G _dc is greater than 𝛹_pm∕𝛹_dc until a _p is equal to 3.2 m, which means that the maximum extension length l _max in the mudstone layer is 3.2 m. Compared with the monitored data in the field of the southern Qinshui Basin (the average length is 2 m, Zhang et al. [20]), the error between them may be caused by the assumption of perfect interface between coal seam layer and mudstone layer. It is proved that the FEM model built in this paper is reliable. Some factors that influenced the behavior of vertical fracture penetration/deflection at the interface are investigated in the following parts.

5.2. The influence of interfacial stiffness on the fracture propagation behavior

It is very important to analyse the influence of the interfacial effect on the propagation behavior of vertical fracture. Usually, the interfacial stiffness is used to represent the interfacial effect. Based on this, some different types of interface can be divided with the different interfacial stiffness, such as perfect interface with the interfacial stiffness 𝛽_yy → ∞ and 𝛽_xy → ∞, separation interface with the interfacial stiffness 𝛽_yy → 0 and 𝛽_xy → 0, general case with the interfacial stiffness 𝛽_yy and 𝛽_xy being the finite value. Usually, the values of the interfacial stiffness can be listed as 𝛽_yy = 𝛽_xy = 10ⁿ (n = 0,1,2,3, …14). For convenience, the exponential n is used to represent the interfacial stiffness. With the different interfacial stiffness, the propagation behaviors of vertical fracture at the interface are carried out.

Figure 9 shows the relationship between interfacial stiffness and G _pm∕G _dc. With the increase of interfacial stiffness, the trend of G _pm∕G _dc change can be divided into three stages: the beginning stage in which the interfacial stiffness is smaller than 10⁹ Pa/m (Fig. 11 shows the shear slide at the interface with this situation); the linear increase stage in which the interfacial stiffness is greater than 10⁹ Pa/m and smaller than 10¹² Pa/m; and the final stable stage in which the interfacial stiffness is greater than 10¹² Pa/m. Accordingly, the change law of the maximum extension length l _max is also divided into three stages, as is shown in Fig. 9. Meanwhile, the maximum extension length l _max is 2 m, which is in agreement with the field monitoring results when the interfacial stiffness equals 10¹⁰ Pa. It means that the interface between coal seam layer and mudstone layer is not perfect and its value is approximately equal to 10¹⁰ Pa. In the following study, the interfacial stiffness is set as 10¹⁰ Pa.

Fig. 9.

The relationship between interfacial stiffness n and G _pm∕G _dc.

Fig. 10.

The relationship between interfacial stiffness n and l _max.

Fig. 11.

The shear slide at the interface with the interfacial stiffness n smaller than 10⁹ Pa/m.

5.3. The influence of thickness of coal seam layer on the fracture propagation behavior

The thickness of coal seam in coalbed methane wells in the southern Qinshui Basin is diverse, approximately 6--8 m. Figure 12 shows the relationship between the half thicknesses of coal seam h _cm (2, 2.5, 3, 3.5 and 4 m) and the maximum extension length l _max. As the h _cm increases, the length of l _max slowly decreases.

Fig. 12.

The influence of half thickness of coal seam layer h _cm on the maximum extension length l _max.

5.4. The influence of Young’s modulus of sandstone cap on the fracture propagation behavior

The No. 3 coal seam layer and sandstone layer in the southern Qinshui Basin are usually capped by sandstone. The average Young’s modulus of sandstone cap E _m in the area is about 34 GPa. Figure 13 exhibits the change law of the maximum extension length l _max with the different E _m (28 GPa, 31 GPa, 34 GPa, 37 GPa and 40 GPa). With the increase of E _m, the maximum extension length l _max decreases gradually. It is shown that incresing E _m contributes to the cease of the fracture propagation in the mudstone layer.

Fig. 13.

The influence of Young’s modulus of sandstone cap E _m on the maximum extension length l _max.

5.5. The influence of in-situ stress on the fracture propagation behavior

Influenced by geological structure and lithology, the in-situ stress between coal seam and mudstone layers is different. In this study, the maximum extension length l _max of the vertical fracture in mudstone at different horizontal in-situ stress differences Δ𝜎 (0.2 MPa, 0.4 MPa, 0.6 MPa, 0.8 MPa and 1 MPa) are calculated. As is shown in Fig. 14, a positive relationship of 𝛥𝜎 and l _max means that the Δ𝜎 has great influence on the l _max in mudstone layer.

Fig. 14.

The influence of horizontal in-situ stress difference Δ𝜎 on the maximum extension length l _max.

6. Conclusion

(1) According to energy release rate criterion, using linear spring model to simulate the interfacial effect is proved to be an effective method to study the behavior of vertical fracture penetration/deflection at the interface between coal seam and mudstone layers.

(2) The interfacial stiffness between coal seam and mudstone layers varies between 10⁹ Pa/m and 10¹² Pa/m (average is 10¹⁰ Pa/m). When the interfacial stiffness is less than 10⁹ Pa/m or greater than 10¹² Pa/m, the interface is in a separate or perfect state.

(3) With the increase of Young’s modulus of sandstone cap, the maximum extension length in mudstone decreases gradually suggesting that Young’s modulus of sandstone cap has an obvious effect on stopping the fracture propagation.

(4) The horizontal in-situ stress difference is an important influence on the vertical fracture propagation.

Footnotes

Acknowledgements

This project is financially supported by the National Natural Science Foundation of China (Grant No. 51704037). The authors highly appreciate the anonymous reviewers for their constructive comments and suggestions.

Conflict of interest

None to report.

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